1549380232-Automorphic_Forms_and_Applications__Sarnak_

188 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £ -FUNCTIONS

where q11: ~ 1 is an integer (the arithmetic conductor of n) supported at

the finite ramified places for n, w(n) is a complex number of modulus 1

(the root number) and ii" denotes the contragredient of n. In particular,

one has qir = q11:, and for any place v L v(ir, s) = L v (n, s) .)

• L = (n, s)L(n , s ) is bounded in vertical strips (and in fact has exponential
  decay) and is of finite order away from its poles (if any).
  It is convenient to encapsulate the main parameters attached to n in a single quan-
  tity that occurs in many problems as a normalizing factor; for that purpose, Iwaniec
  and Sarnak [IS2] introduced the analytic conductor of n: it is a function over the
  reals given by
  d
  t E R-> Q11:(t) = q11: II(l +lit-μ11:,il) = : q11:Q11: 00 (t ).
  i=l
  For the rest of these lectures, we denote Q 11: ( 0) by Q 11:.

Remark 1.1. We mostly concentrate on cuspidal £-functions because they form

the building blocks for £-functions of general automorphic representations. Indeed,

given d 1 ,... , dr with d 1 +· · ·+dr = d, the Langlands theory of Eisenstein series asso-

ciates to an r -tuple ( n 1 , ... , n r ) of (not necessarily unitary) cuspidal representations

a distinguished automorphic representation of G L d ( AQ), denoted n = n 1 EB· · · EBn n

the isobaric sum of the ni, i = 1 ... , r. By construction, cuspidal representations

are isobaric and it is a result of Shalika that the ni appearing in the construction

of n (the constituents of n ) are unique up to permutation (see [Co2]). Then the

£-function of n is given by the product

r

L(n1 EB· · · EB 'Irr, s) =II L (ni, s).

i=l

Langlands also proved that any automorphic representation n is nearly equivalent

to an isobaric sum n ' (i.e. for almost every place v, 'Irv ~ n~), and as a consequence

L(n , s) and L (n' , s) coincide up to finitely many local factors.

1.1.2. £-functions of pairs

Another class of £ -functions fundamental to the whole theory are the Rankin/ -
Selberg type £-functions L ( n 0 n' , s ) associated to pairs of automorphic repre-
sentations (n, n' ) E A~(Q) x A~,(Q); their theory was initiated by Rankin and
Selberg in the case of classical modular forms [Ran, Se]. For general automor-
phic forms, the analytic theory of £-functions of pairs was initiated and devel-
opped in several papers by Jacquet, Piatetsky-Shapiro and Shalika [J, JS2, JPSS2]
and completed in works of Shahidi, Moeglin/ Waldspurger and Gelbart/ Shahidi
[Shal, Sha2, Sha3, MWl, GeSh]; we refer again to [Co2] for a detailled ex-
position of their construction and the derivation of their basic properties. Given
(n, n') E A~(Q) x A^0 (d'), the Rankin/ Selberg type £ -function L(n 0 n ' , s ) is a
Dirichlet series

L (n 0 n ' , s) =II L p(n 0 n', s) = L A11:®:~(n),

P n~l